A self-duality of strong starlikeness

In this note, we will show that a simply connected bounded domain D is strongly starlike of order α < 1 with respect to the origin if and only if so is D˅, where D˅ is the analytic inversion of the exterior of D, namely, D˅ = {w ∈ C: 1/w ∈ Ĉ\˅}. This fact neatly explains the relationship between some known properties of strongly starlike domains and provides several new characterizations for those domains.